1. **State the problem:** We are given the line equation $x + 4y = 8$ and need to find the slopes of lines perpendicular and parallel to it. 2. **Rewrite the line in slope-intercept form:** The slope-intercept form is $y = mx + b$, where $m$ is the slope. Starting with: $$x + 4y = 8$$ Subtract $x$ from both sides: $$4y = -x + 8$$ Divide both sides by 4: $$y = \frac{\cancel{4}y}{\cancel{4}} = \frac{-x + 8}{4} = -\frac{1}{4}x + 2$$ 3. **Identify the slope of the given line:** From the equation $y = -\frac{1}{4}x + 2$, the slope $m$ is $-\frac{1}{4}$. 4. **Find the slope of a line perpendicular:** The slope of a line perpendicular to another is the negative reciprocal of the original slope. Calculate the negative reciprocal: $$m_{\perp} = -\frac{1}{m} = -\frac{1}{-\frac{1}{4}} = 4$$ 5. **Find the slope of a line parallel:** The slope of a line parallel to another is the same as the original slope. So, $$m_{\parallel} = -\frac{1}{4}$$ **Final answers:** - Slope of a perpendicular line: $4$ - Slope of a parallel line: $-\frac{1}{4}$